#8602781

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,7)(3,9)(8,11)(10,15)(17,18)(19,24)(20,21)(22,27)(23,25)(26,32)(28,29)(30,31)(33,34,40,45)(35,38,46,41)(36,43,37,42)(39,44,47,48);(1,2,6,7)(3,5,9,14)(4,11,12,8)(10,16,15,13)(17,19)(18,24)(20,30)(21,31)(22,25)(23,27)(26,28)(29,32)(33,35,40,46)(34,41,45,38)(36,39,37,47)(42,44,43,48);(4,12)(8,11)(10,15)(13,16)(18,25)(19,27)(21,32)(28,30)(33,36,40,37)(34,42,45,43)(35,39,46,47)(38,48,41,44);(1,3)(2,5)(4,10)(6,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,26)(18,21,25,32)(19,28,27,30)(22,31,24,29)(33,37,40,36)(34,43,45,42)(35,39,46,47)(38,48,41,44);(1,4,6,12)(2,8,7,11)(3,10,9,15)(5,13,14,16)(17,21)(18,26)(19,29)(20,25)(22,28)(23,32)(24,30)(27,31)(33,38,40,41)(34,35,45,46)(36,44,37,48)(39,42,47,43);(1,5)(2,9)(3,7)(4,13)(6,14)(8,15)(10,11)(12,16)(17,22)(18,27)(19,25)(20,29)(21,28)(23,24)(26,31)(30,32)(33,39)(34,44)(35,37)(36,46)(38,42)(40,47)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8509977);